Is this a covering space? - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-25T00:41:51Z http://mathoverflow.net/feeds/question/29213 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/29213/is-this-a-covering-space Is this a covering space? Changwei Zhou 2010-06-23T10:27:27Z 2010-06-23T10:46:41Z <p>In Hatcher's book Page 79 I was asked to provide two-sheeted covering space $Y \rightarrow X_{1}$ such that the composition $Y \rightarrow X_{1} \rightarrow X$ of the two covering spaces is not a covering space. In here the space $X$ is the shrinking wedge of circles, and $X_{1}$ is placing infinite such spaces onto the line. </p> <p>(See the figure in the book) <a href="http://www.math.cornell.edu/~hatcher/AT/ATpage.html" rel="nofollow">http://www.math.cornell.edu/~hatcher/AT/ATpage.html</a></p> <p>The example I imagined is this one:</p> <p>I use $Y$ the same as $X_{1}$, but I make the mapping $Y\rightarrow X_{1}$ like this: I map and the second circle to the first circle, and map the rest to themselves. Then if $Y \rightarrow X_{1} \rightarrow X$ is a covering map, according to the defintion the inverse of a neighborhood in $X$, they must be disjoint; but in here they simply coincide. </p> <p>I don't know whether this really works as he required. Mostly because the original space is sufficiently bad (not locally path connected) therefore I expect Hatcher would need me to utilize this condition. Also, I want to ask if one can assert that if $X$ is locally path connected, then $Y$ must be a covering space of X. I'm thinking about this because in the next page problem 16, Hatcher asked the reader to prove this:</p> <p>"16. Give maps $X\rightarrow Y\rightarrow Z$ such that both $Y\rightarrow Z$ and the composition $X\rightarrow Z$ are covering spaces, show that $X\rightarrow Y$ is a covering space if $Z$ is locally path-connected...."</p> <p>Sorry that this is a purely homework level question. </p>